We investigate the precise large deviations for random sums of extended negatively dependent random variables with long and dominatedly varying tails. We find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence. We apply the results to a generalized dependent compound renewal risk model including premium process and claim process and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.
The study of large deviations plays an important role in insurance and finance theory. The aim of this paper is to study precise large deviation probabilities for sequences of dependent and heavy-tailed random variables. Let
Some earlier work, for the case
One of the main concepts we use is the extended negative dependence, which was first introduced by Liu [
We call random variables
Recall that if
In presence of END structure, Liu [
The basic assumption of this paper is that
The rest of the paper is organized as follows. After simply reviewing some subclasses of heavy-tailed distributions and giving some lemmas needed to prove the theorem in Section
To formulate our main results we need to introduce some notations and assumptions. Throughout this paper we restrict ourselves to the case that
A distribution function (d.f.)
A d.f.
A d.f.
It is well known that the previous heavy-tailed distribution classes have the following relationship:
For more details about heavy-tailed class in the context of insurance and finance, we refer the reader to Embrechts et al. [
For a d.f.
We close this section by explaining some symbols which will be used later. We will use
We will need some lemmas used in the proofs of our theorems. The following lemma is given by Tang [
If for each it holds for each
If
The proof is analogous to that of Lemma 2.2 of Chen and Zhang [
By Definition
Let for any
Wang et al. [
Let
Let
Let
(i) Consider
By
(ii) Consider
First we shows that
Let
Firstly, by Lemma
Let
This lemma plays an important role in the proof of Theorem
Let
From the following proof, it is easy to see that (
Under the assumptions of Theorem
From Remark
In the sequel,
In order to prove Theorem
By Tang et al. [
Therefore, Theorem
Assume that Assumptions
In order to prove this lemma, we consider the following three cases.
(i)
Choose
(ii)
Choose
(iii)
Let
For
Assume that Assumptions
Similar to the proof of Lemma
(i)
Firstly, for
(ii)
Similarly, for any
(iii)
Since
Assume that Assumptions
We prove this lemma by splitting into three cases like the former lemmas’ proof.
(i)
For
(ii)
Similarly, by Lemma
(iii)
Note that
From a realistic point of view, we further generalize such work to a much more realistic model including premium income process. The premium income process depends not only on the number of customers who buy the insurance portfolios but also on the premium size process. The risk model has the following structure. The individual claim sizes The number of claims in the interval The number of customers who buy the insurance portfolios within the time interval The premium size process
Suppose that
For the generalized dependent compound renewal risk model (
Applying Theorem
By the same argument as in Theorem
Under the conditions of Theorem
If we assume that
From the previous two theorems, we find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence.
Observing that
We proceed a series of lemmas to prove Theorem
For
For
For
For any
Again using Theorem
Hence, combining (
For
For
The first author is supported by National Science Foundation of China (nos. 10801124 and 11171321) and the Fundamental Research Funds for the Central Universities (no. WK 2040170006).